Wessel was born in Bielefeld in Westphalia, the son of Dr. Ludwig Wessel, a Lutheran minister at the Nikolaikirche, one of Berlin's oldest churches. Wessel's mother, Luise Margarete Wessel, also came from a family of Lutheran pastors. The family lived in the nearby Judenstraße (the Jews' Street), which in mediaeval times had been the centre of Berlin's Jewish community. Wessel's father was a supporter of the conservative German National People's Party (DNVP), and when he was 15, Wessel joined the DNVP youth group, the Bismarckjugend. He soon became a local leader, engaging in street battles with youth members of the Social Democratic Party and Communist Party.
Wessel attended the Volksschule (primary school) of Köllnisches Gymnasium from 1914 to 1922, thereafter the Königstädtisches Gymnasium (high school) and for his final year of school the Luisenstädtisches Gymnasium, where he passed his Abitur examination. In April 1926 he enrolled in the law faculty of Friedrich Wilhelm University Unter den Linden.